Module Theory

This expository monograph was written for three reasons. Firstly we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the \Krull-Schmidt Theorem\ holds for ar­ tinian modules. The problem remained open for 63 years: its solution a negative answer to Krulls question was published only in 1995 (see [Facchini Herbera Levy and Vamos]). Secondly we wanted to present the answer to a question posed by Warfield in 1975 [Warfield 75]. He proved that every finitely pre­ sented module over a serial ring is a direct sum of uniserial modules and asked if such a decomposition was unique. In other words Warfield asked whether the \Krull-Schmidt Theorem\ holds for serial modules. The solution to this problem a negative answer again appeared in [Facchini 96]. Thirdly the so­ lution to Warfields problem shows interesting behavior a rare phenomenon in the history of Krull-Schmidt type theorems. Essentially the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold any two indecomposable decompositions are uniquely determined up to a permutation and when it does not hold for a class of modules this is proved via an example. For serial modules the Krull-Schmidt Theorem does not hold but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phenomenon to a wider math­ ematical audience.